Course subject: Applied Mathematics (AMATH)

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Applied Mathematics (AMATH) 642 Computational Methods for Partial Differential Equations (0.50) LEC

Course ID: 016245
This course studies several classes of methods for the numerical solution of partial differential equations in multiple dimensions on structured and unstructured grids. Finite volume methods for hyperbolic conservation laws: linear and nonlinear hyperbolic systems; stability; numerical conservation. Finite element methods for elliptic and parabolic equations: weak forms; existence of solutions; optimal convergence; higher-order methods. Examples from fluid and solid mechanics. Additional topics as time permits.

Applied Mathematics (AMATH) 651 Introduction to Dynamical Systems (0.50) LEC

Course ID: 016207
A unified view of linear and nonlinear systems of ordinary differential equations in Rn. Flow operators and their classification: contractions, expansions, hyperbolic flows. Stable and unstable manifolds. Phase-space analysis. Nonlinear systems, stability of equilibria and Lyapunov functions. The special case of flows in the plane, Poincare-Bendixson theorem and limit cycles. Applications to physical problems will be a motivating influence. (Heldwith AMATH 451)

Applied Mathematics (AMATH) 653 Partial Differential Equations 2 (0.50) LEC

Course ID: 016208
A thorough discussion of the class of second order linear partial differential equations with constant coefficients, in two independent variables. Laplace's equation, the wave equation and the heat equation in higher dimensions. Theoretical/qualitative aspects: well-posed problems, maximum principles for elliptic and parabolic equations, continuous dependence results, uniqueness results (including consideration of unbounded domains), domain of dependence for hyperbolic equations. Solution procedures: elliptic equations -- Green functions, conformal mapping; hyperbolic equations -- generalized d'Alembert solution, spherical means, method of descent; transform methods -- Fourier, multiple Fourier, Laplace, Hankel (for all three types of partial differential equations); Duhamel's method for inhomogeneous hyperbolic and parabolic equations. (Heldwith AMATH 453)

Applied Mathematics (AMATH) 655 Control Theory (0.50) LEC

Course ID: 011278
Feedback control with applications. System theory in both time and frequency domain, state-space computations, stability, system uncertainty, loopshaping, linear quadratic regulators and estimation. (Heldwith AMATH 455)

Applied Mathematics (AMATH) 656 Calculus of Variations (0.50) LEC

Course ID: 016209
Concept of functional and its variations. The solution of problems using variational methods - the Euler- Lagrange equations. Applications include an introduction to Hamilton's principle and optimal control. (Heldwith AMATH 456)

Applied Mathematics (AMATH) 663 Fluid Mechanics (0.50) LEC

Course ID: 011279
Incompressible, irrotational flow. Incompressible viscous flow. Introduction to wave motion and geophysical fluid mechanics. Elements of compressible flow. (Heldwith AMATH 463)

Applied Mathematics (AMATH) 673 Quantum Theory 2 (0.50) LEC

Course ID: 011280
The Hilbert space of states, observables, and time evolution. Feynman path integral and Greens functions. Approximation methods. Co-ordinate transformations, angular momentum, and spin. The relation between symmetries and conservation laws. Density matrix, Ehrenfest theorem, and decoherence. Multiparticle quantum mechanics. Bell inequality and basics of quantum computing.

Applied Mathematics (AMATH) 674 Quantum Theory 3: Quantum Information and Foundations (0.50) LEC

Course ID: 016118
Theory of correlations and entanglement, theory of quantum channels, detectors and the measurement problem in quantum mechanics, phase space formulation of quantum mechanics and entanglement in infinite dimensional quantum systems, introduction to open quantum systems, exploration of current research directions in quantum information.

Applied Mathematics (AMATH) 675 Introduction to General Relativity (0.50) LEC

Course ID: 011281
Tensor analysis. Curved space-time and the Einstein field equations. The Schwarzschild solution and applications. The Friedmann-Robertson-Walker cosmological models.

Applied Mathematics (AMATH) 677 Stochastic Processes for Applied Mathematics (0.50) LEC

Course ID: 016210
Random variables, expectations, conditional probabilities, conditional expectations, convergence of a sequence of random variables, limit theorems, minimum mean square error estimation, the orthogonality principle, random process, discrete-time and continuous-time Markov chains and applications, forward and backward equation, invariant distribution, Gaussian process and Brownian motion, expectation maximization algorithm, linear discrete stochastic equations, linear innovation sequences, Kalman filter, various applications. (Heldwith AMATH 477)

Applied Mathematics (AMATH) 731 Applied Functional Analysis (0.50) LEC

Course ID: 000116
Basic concepts of functional analysis. Topics include: theory of linear operators, nonlinear operators and the Frechet derivative, fixed point theorems, approximate solution of operator equations, Hilbert space, spectral theory. Applications from various areas will be used to motivate and illustrate the theory. A previous undergraduate course in real analysis is strongly recommended.

Applied Mathematics (AMATH) 732 Asymptotic Analysis and Perturbation Theory (0.50) LEC

Course ID: 011283
Elements of asymptotic analysis. Techniques of perturbation theory such as Poincare-Lindstedt, matched asymptotic expansions and multiple scales. Applications to various areas form an essential aspect of the course. Previous courses in real analysis and differential equations at the undergraduate level are strongly recommended.

Applied Mathematics (AMATH) 740 Numerical Analysis (0.50) LEC

Course ID: 012670
Introduction to basic algorithms and techniques for numerical computing. Error analysis, interpolation (including splines), numerical differentiation and integration, numerical linear algebra (including methods for linear systems, eigenvalue problems, and the singular value decomposition), root finding for nonlinear equations and systems, numerical ordinary differential equations, and approximation methods (including least squares, orthogonal polynomials, and Fourier transforms).

Applied Mathematics (AMATH) 741 Numerical Solution of Partial Differential Equations (0.50) LEC

Course ID: 000724
Discretization methods for partial differential equations, including finite difference, finite volume and finite element methods. Application to elliptic, hyperbolic and parabolic equations. Convergence and stability issues, properties of discrete equations, and treatment of non-linearities. Stiffness matrix assembly and use of sparse matric software. Students should have completed a course in numerical computation at the undergraduate level.

Applied Mathematics (AMATH) 751 Advanced Ordinary Differential Equations (0.50) LEC

Course ID: 000118
Qualitative theory of systems of ODEs. Topics include: existence/uniqueness of solutions, comparison principle, iterative techniques, stability and boundedness, Lyapunov method, periodic solutions, Floquet theory and Poincare maps, hyperbolicity, stable, unstable and center manifolds, structural stability and bifurcation. Applications from various areas will be used to motivate and illustrate the theory. A previous course in ordinary differential equations at the undergraduate level is strongly recommended.

Applied Mathematics (AMATH) 753 Advanced Partial Differential Equations (0.50) LEC

Course ID: 000119
The main themes are well-posedness of problems, Hilbert space methods, variational principles and integral equation methods. Topics include: first-order nonlinear partial differential equations, quasilinear hyperbolic systems, potential theory, eigenfunctions and eigenvalues, semi-groups, and power series solutions. Applications from various areas will be used to motivate and illustrate the theory. A previous course in partial differential equations at the undergraduate level is strongly recommended.

Applied Mathematics (AMATH) 777 Stochastic Processes in the Physical Sciences (0.50) LEC

Course ID: 011284
Basic concepts and classification of stochastic processes. Stochastic differentiation and integration, Markov processes, Chapman-Kolmogorov equation, Fokker-Planck equation, Master equations: mesoscopic vs. macroscopic description. Spectral representation of stationary processes. Correlation function theory. A previous course in probability theory at the undergraduate level is strongly recommended.

Applied Mathematics (AMATH) 840 Advanced Numerical Methods for Computational and Data Sciences (0.50) LEC

Course ID: 016229
Theory and practice of a selection of advanced numerical methods for computational and data sciences. Algorithms for eigenvalues and singular value decomposition. Multigrid methods for linear and nonlinear systems. Sparse optimization and compressed sensing. Low-rank tensor and matrix decomposition. Nonlinear convergence acceleration. Randomized numerical linear algebra. Adjoint methods and automatic differentiation for neural networks and optimal control. Stochastic gradient descent and variants. Efficient computer implementation of the algorithms and applications with real-world data. Students should have completed an introductory course on numerical methods.

Applied Mathematics (AMATH) 841 Finite Element Methods (0.50) LEC

Course ID: 016230
Mathematical theory of finite element methods, with error analysis and stability. Continuous and discontinuous Galerkin methods and mixed finite element methods, applied to elliptic, parabolic, and hyperbolic partial differential equations including compressible and/or incompressible Euler and Navier-Stokes equations. Implementation in two and three dimensions. Students should have completed an introductory course on numerical methods for partial differential equations.

Applied Mathematics (AMATH) 851 Stability Theory and Applications (0.50) LEC

Course ID: 011285
Concepts of stability and boundedness, basic stability criteria, comparison methods, large scale systems, method of decomposition and aggregation, method of several Lyapunov functions, method of vector Lyapunov functions, method of higher derivatives. Stability problems in ecology, mechanics, neural networks and control systems. Students should have completed AM751 or equivalent.

Applied Mathematics (AMATH) 855 Advanced Systems Analysis and Control (0.50) LEC

Course ID: 000132
The main theme is the extension of control theory beyond systems modelled by linear ordinary differential equations. Topics include: advanced systems theory, control of nonlinear systems, control of partial differential equations and delay equations. Students should have completed an introductory undergraduate course in control theory.

Applied Mathematics (AMATH) 863 Hydrodynamic Stability and Turbulence (0.50) LEC

Course ID: 000133
Mathematical methods, stability of parellel flows for unstratified and stratified fluids, Rayleigh-Taylor instability, centrifugal instability, barotropic and baroclinic instabilities, the effects of viscosity and the Orr-Sommerfeld equation, transition to turbulence, averaged equations, closure problem, homogeneous isotropic turbulence, turbulent boundary layers, effects of stratification. Students should have completed an introductory undergraduate course in fluid mechanics.

Applied Mathematics (AMATH) 867 Dispersive and Nonlinear Waves (0.50) LEC

Course ID: 000134
Dispersive waves, propagation of dispersive waves in an inhomogeneous medium (WKB theory). Nonlinear resonant interactions. Solitons: completely integrable nonlinear wave equations (e.g., the KdV equation, nonlinear Schrodinger equations) and the inverse Scattering Transform. Applications to water waves and nonlinear optics. Introducation to weakly nonlocal solitary waves and beyond-all-orders asymptotics. Completion of an upper year course in partial differential equations is strongly recommended.

Applied Mathematics (AMATH) 871 Quantum Information Processing (0.50) LEC

Course ID: 011589
Review of basics of quantum information and computational complexity; Simple quantum algorithms; Quantum Fourier transform and Shor factoring algorithm: Amplitude amplification, Grover search algorithm and its optimality; Completely positive trace-preserving maps and Kraus representation; Non-locality and communication complexity; Physical realizations of quantum computation: requirements and examples; Quantum error-correction, including CSS codes, and elements of fault-tolerant computation; Quantum cryptography; Security proofs of quantum key distribution protocols; Quantum proof systems. Familiarity with theoretical computer science or quantum mechanics will also be an asset, though most students will not be familiar with both.

Applied Mathematics (AMATH) 872 Introduction to Quantum Field Theory for Cosmology (0.50) LEC

Course ID: 012151
Introduction to scalar field theory and its canonical quantization in flat and curved spacetimes. The flat space effects of Casimir and Unruh. Quantum fluctuations of scalar fields and of the metric on curved space-times and application to inflationary cosmology. Hawking radiation.

Applied Mathematics (AMATH) 873 Introduction to Quantum Field Theory (0.50) LEC

Course ID: 000135
Review of relativistic quantum mechanics and classical field theory. Quantization of free quantum fields (the particle interpretation of field quanta). Canonical quantization of interacting fields (Feynman rules). Application of the formalism of interacting quantum fields to lowest¿order quantum electrodynamic processes. Radiative corrections and renormalization.

Applied Mathematics (AMATH) 874 Advanced techniques in General Relativity and Applications to Black Holes (0.50) LEC

Course ID: 010439
Review of elementary general relativity. Timelike and null geodesic congruences. Hypersurfaces and junction conditions. Lagrangian and Hamiltonian formulations of general relativity. Mass and angular momentum of a gravitating body. The laws of black-hole mechanics.

Applied Mathematics (AMATH) 875 Introduction to General Relativity with Applications to Cosmology (0.50) LEC

Course ID: 011282
Introduction to the differential geometry of Lorentzian manifolds. The priniciples of general relativity. Causal structure and cosmological singularities. Cosmological space-times with Killing vector fields. Friedmann-Lemaitre cosmologies, scalar, vector and tensor perburbations in the linear and nonlinear regimes. De Sitter space-times and inflationary models.

Applied Mathematics (AMATH) 876 Open Quantum Systems (0.50) LEC

Course ID: 012567
Review of the axioms of quantum theory and derivation of generalized axioms by considering states, transformations, and measurements in an extended Hilbert space. Master equations and the Markov approximation. Standard models of system-environment interactions and the phenomenology of decoherence. Introduction to quantum control with applications in NMR, quantum optics, and quantum computing.

Applied Mathematics (AMATH) 877 Foundations of Quantum Theory (0.50) LEC

Course ID: 000136
Review mathematical formulation of operational quantum theory; theory of measurements and decoherence; quantum-classical contrast; review of historical perspectives on interpretation, including EPR paradox; Bell's theorem, non-locality and contextuality; PBR theorem; selected topics including overviews of current interpretations of quantum mechanics and critical experiments in quantum foundations.

Applied Mathematics (AMATH) 881 Introduction to Mathematical Oncology (0.50) LEC

Course ID: 013468
Biological and clinical aspects of cancer. Overview of recent mathematical models developed to examine different stages of cancer growth and therapeutic strategies, including ordinary and partial differential equation models, discrete models and models based on continuum mechanics. Various analytical and numerical methods will be used in the analysis of these models.

Applied Mathematics (AMATH) 882 Mathematical Cell Biology (0.50) LEC

Course ID: 013469
Dynamic mathematical modelling of biological process at the cellular level. Intracellular networks: metabolism, signal transduction, and genetic regulatory networks. Neural networks: from biophysical modelling of single neurons to the analysis of network behaviour. Modelling will be carried out primarily through ordinary differential equations; analysis will involve application of dynamical systems tools and simulations. Other relevant modelling frameworks (PDEs, delay equations, stochastic methods) will be touched on as time allows.